Hello all, We are starting Banach spaces webinars for this academic year on Friday (8/27) with the following talk. Looking forward to seeing you all back. Best regards, Bunyamin For past talks and videos please see the webinar website: http://www.math.unt.edu/~bunyamin/banach ******** Speaker: Willian Corrêa (Universidade de São Paulo) Title: Two steps into the homology of $\ell_2$ Abstract. Enflo, Lindenstrauss and Pisier showed in 1975 the existence of a short exact sequence $0 \rightarrow \ell_2 \rightarrow X \rightarrow \ell_2 \rightarrow 0$ of Banach spaces and bounded linear operators in which $\ell_2$ is not complemented in $X$, i.e., $X \neq \ell_2$. This means that there is a non-Hilbertian Banach space with an isomorphic copy of $\ell_2$ such that the respective quotient is isomorphic to $\ell_2$ as well. In homological language, they showed that $Ext(\ell_2, \ell_2) \neq 0$. In this talk we discuss the next level of homology, i.e., we study exact sequences $0 \rightarrow \ell_2 \rightarrow X_1 \rightarrow X_2 \rightarrow \ell_2 \rightarrow 0$ and present the recent result that $Ext^2(\ell_2, \ell_2) \neq 0$. Joint work with Félix Cabello Sánchez, Jesús M. F. Castillo, Valentin Ferenczi and Ricardo García. The author was supported by FAPESP, processes 2016/25574-8 and 2018/03765-1.